Relevance of the small frequency separation for asteroseismic stellar age, mass, and radius: A statistical investigation for main-sequence low-mass stars

Aims. We performed a theoretical analysis aimed at quantifying the relevance of the small frequency separation delta nu in determining stellar ages, masses, and radii. We aimed to establish a minimum uncertainty on these quantities for low-mass stars across different evolutionary stages of the main sequence and to evaluate the biases that come from some systematic differences between the stellar model grid adopted for the recovery and the observed stars.Methods. We adopted the Stellar CharactEristics Pisa Estimation gRid (SCEPtER) pipeline for low-mass stars, [0.7, 1.05] M-circle dot, from the zero-age main sequence (ZAMS) to the central hydrogen depletion. For each model in the grid, we computed oscillation frequencies. Synthetic stars were generated and reconstructed based on different assumptions about the relative precision in the delta nu parameter (namely 5% and 2%). The quantification of the systematic errors arising from a possible mismatch between synthetic stars and the recovery grid was performed by generating stars from synthetic grids of stellar models with different initial helium abundance and microscopic diffusion efficiency. The results obtained without delta nu as an observable are included for comparison.Results. The investigation highlighted and confirmed the improvement in the age estimates when delta nu is available, which has already been reported in the literature. While the biases were negligible, the statistical error affecting age estimates was strongly dependent on the stellar evolutionary phase. The error is at its maximum at ZAMS and it decreases to about 11% and 6% (delta nu known at 5% and 2% level, respectively) when stars reach the 30% of their evolutionary MS lifetime. The usefulness of small frequency separation in improving age estimates vanishes in the last 20% of the MS. The availability of delta nu in the fit for mass and radius estimates provided an effect that was nearly identical to its effect on age, assuming an observational uncertainty of 5%. As a departure, with respect to age estimates, no benefit was detected for mass and radius determinations from a reduction of the observational error in delta nu to 2%. The age variability attributed to differences in the initial helium abundance resulted in negligible results owing to compensation effects that have already been discussed in previous works. On the other hand, the current uncertainty in the initial helium abundance leads to a greater bias (2% and 1% level) in mass and radius estimates whenever delta nu is in the observational pool. This result, together with the presence of further unexplored uncertainty sources, suggest that precision in the derived stellar quantities below these thresholds may possibly be overoptimistic. The impact of microscopic diffusion was investigated by adopting a grid of models for the recovery which totally neglected the process. The availability of the small frequency separation resulted in biases lower than 5% and 2% for observational errors of 5% and 2%, respectively. The estimates of mass and radius showed again a greater distortion when delta nu is included among the observables. These biases are at the level of 1%, confirming that threshold as a minimum realistic uncertainty on the derived stellar quantities. Finally, we compared the estimates by the SCEPtER pipeline for 13 Kepler asteroseismic LEGACY sample stars with those given by six different pipelines from literature. This procedure demonstrated a fair agreement for the results. The comparison suggests that a realistic approach to the determination of the error on the estimated parameters consists of approximately doubling the error in the recovered stellar characteristics from a single pipeline. Overall, on the LEGACY sample data, we obtained a multi-pipeline precision of about 4.4%, 1.7%, and 11% on the estimated masses, radii, and ages, respectively.